Fig. 2

Characterization of encoded states. a U_enc and U_dec are operations that coherently map between two distinct two-dimensional subspaces, represented by Bloch spheres. The first subspace consists of the transmon \(\left| g \right\rangle \) and \(\left| e \right\rangle \) levels, with the oscillator in the vacuum. The second is given by the oscillator-encoded states \(\left| { + {{\rm{Z}}_{\rm{L}}}} \right\rangle \) and \(\left| { - {{\rm{Z}}_{\rm{L}}}} \right\rangle \) (Eq. (1)), with the transmon in the ground state. b Wigner tomography sequence that characterizes the encoded states. A transmon state is prepared by applying an initial rotation U_i and is mapped to the oscillator using U_enc. An oscillator displacement D _β followed by a parity mapping operation Π (implemented using an optimal control pulse) allows one to measure the oscillator Wigner function W(β). The transmon can be re-used to measure the oscillator’s parity because the encoding pulse leaves the transmon in the ground state with high probability (p>98%). c Applying U_enc to the transmon states \(\left| g \right\rangle \) and \(\left| e \right\rangle \) produces states whose Wigner functions are consistent with the intended encoded basis states (Eq. (1)). A transmon spectroscopy experiment (top panel) illustrates that only photon number states with n = 0 mod 4 and n = 2 mod 4 are present for logical state \(\left| { + {{\rm{Z}}_{\rm{L}}}} \right\rangle \) and \(\left| { - {{\rm{Z}}_{\rm{L}}}} \right\rangle \) _, respectively. d Applying U_enc to superpositions of the transmon basis states demonstrates that the relative phase is preserved and that U_enc is a faithful map between the transmon and logical qubit Bloch spheres. These states, on the equator of the Bloch sphere, are equally weighted superpositions of \(\left| { + {{\rm{Z}}_{\rm{L}}}} \right\rangle \) and \(\left| { - {{\rm{Z}}_{\rm{L}}}} \right\rangle \) and, therefore, contain all even photon numbers present in the basis states